OP-BJPS130058 1..26


Chance, Possibility, and

Explanation
Nina Emery

ABSTRACT

I argue against the common and influential view that non-trivial chances arise only when

the fundamental laws are indeterministic. The problem with this view, I claim, is not that

it conflicts with some antecedently plausible metaphysics of chance or that it fails to

capture our everyday use of ‘chance’ and related terms, but rather that it is unstable.

Any reason for adopting the position that non-trivial chances arise only when the fun-

damental laws are indeterministic is also a reason for adopting a much stronger, and far

less attractive, position. I suggest an alternative account, according to which chances are

probabilities that play a certain explanatory role: they are probabilities that explain

associated frequencies.

1 Introduction

2 A Paradigm Case

3 The Incompatibilist’s Criterion

4 Against the Incompatibilist’s Criterion

5 The Explanatory Criterion

6 Conclusion

1 Introduction

There seems to be an important connection between chance and possibility,

along the following lines: if there is some non-trivial chance that Sally will

attend the party (that is, if the chance of Sally attending the party is between

zero and one), then it is possible for Sally to attend the party and possible for

her to not attend the party. Any theory of chance has a prima facie obligation

to either capture this apparent connection between chance and possibility or

explain it away.

One way of spelling out the connection between chance and possibility is in

terms of the incompatibility of chance and deterministic laws. If there is some

non-trivial chance that Sally will attend the party, then it is possible that Sally

Brit. J. Phil. Sci. 0 (2013), 1–26

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will attend and possible that she will not attend, in the following sense: the

complete physical state of the world right now plus the laws of nature do not

entail that Sally will attend or that she will not attend. Although this view,

which I will call ‘incompatibilism about chance and deterministic laws’, is not

often defended in print, it is widely accepted by philosophers.
1

Arguments

against it (arguments for ‘compatibilism about chance and deterministic

laws’) are taken to establish a significant and surprising result.

In this article, I will argue that this way of thinking is a mistake: compati-

bilism about chance and deterministic laws ought to be the default view. For

one thing, incompatibilism involves taking on substantial metaphysical com-

mitments—commitments that are not necessitated by our best science and that

are, at least on the face of it, rather implausible. For another, there is a

straightforward way to capture the connection between chance and possibility

outlined above without endorsing incompatibilism.

Crucially, my argument does not apply only to those who fall into one or

another of the various partisan camps that have been built up around the

metaphysics of chance; it does not require that you be a Humean, in David

Lewis’s sense, or an advocate of the best systems analysis of laws, or that you

not be a member of either of those groups. (In this way, it differs from argu-

ments for compatibilism put forward by Loewer ([2001]) and Hoefer ([2007]).)

Nor does my argument rely on the assumption that any adequate theory of

chance will capture our everyday use of ‘chance’ and related terms. Although I

take correspondence with ordinary language to be a desirable attribute of any

metaphysical theory, I also leave open the possibility that there may be the-

oretical, scientific, or pragmatic reasons for adopting a philosophical account

of chance in which that correspondence is not perfect. (In this way, my argu-

ment differs from those found in Handfield and Wilson ([forthcoming]) and

Handfield ([2012]).)
2

Instead, my argument begins from two simple assumptions: first, that a

certain case is a paradigm case of chance—that if there is such a thing as

non-trivial chance, it arises in that case; second that a certain widely accepted

theory about the spacetime structure of our world is correct. As I will dem-

onstrate, careful attention to the consequences of that widely accepted theory

shows that any reason for being an incompatibilist about chance and deter-

ministic laws is unstable. Any reason for believing incompatibilism is also a

1
The classic example of this view is found in (Lewis [1980]). A recent example is (Schaffer [2007]).

See (Eagle [2011]) for more on how this debate bears on compatibilism and incompatibilism

regarding determinism and free well.
2

A related assumption, found in (Handfield and Wilson [forthcoming]), as well as in (Glynn

[2010]), which I also avoid, requires that any theory of chance capture our scientific use of

‘chance’ and related terms—in particular, our use of such terms in the non-fundamental

sciences.

Nina Emery2

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reason for believing a stronger claim, and according to that stronger claim,

there are no non-trivial chances, even in the paradigm case.

This argument presents us with a choice: accept the surprising conclusion

that even the paradigm case is not a genuine instance of chance, or reject

incompatibilism. That the latter is the more attractive option is due to the

fact that there is another, often overlooked feature of the paradigm case that

serves as a conceptual constraint on the notion of chance. The details of that

case suggest that chances are distinctive among probabilities in their explana-

tory power. More work remains to be done on this ‘explanatory criterion’, but

even in a minimal and relatively uncontroversial form, it classifies some non-

trivial probabilities that arise when the fundamental laws are deterministic as

genuine instances of chance.

2 A Paradigm Case

Suppose that Sally is a physicist and that she is in the process of running an

experiment that involves sending a stream of silver atoms, which have all been

prepared in a certain way, through a special set of magnets that will deflect

each atom either up or down.
3

The following two facts are true of Sally’s

experimental set-up: (i) the fundamental laws governing the behaviour of

the silver atoms are indeterministic (that is, there is no feature, F, such that

all and only the silver atoms that have F before going through the experimen-

tal set-up will be deflected up)
4

and (ii) the results of the experiment exhibit a

robust pattern: most (but not all) of the silver atoms are deflected up.

This case is generally treated as a paradigm case of chance. Everyone who

thinks that it is possible to identify a distinctively objective sort of probability,

one deserving of the name ‘chance’, thinks that it arises in this sort of case.
5

3
The details do not matter for our purposes, but the sort of case I have in mind is one where the

silver atoms were prepared by being sent through an initial set of Stern-Gerlach magnets, after

which the atoms that were deflected down were discarded, and the experiment consists of

sending the remaining atoms through a second set of magnets, which is rotated, but only

slightly, with respect to the initial set. This sort of experiment is often used to demonstrate

quantum mechanical phenomena and one can find a further description in many introductory

texts on the subject.
4

Whether this is true of the actual world is up for debate. According to the orthodox interpret-

ation of quantum mechanical phenomena (the interpretation presented in physics textbooks) it

is, but other interpretations disagree. Among the contenders taken seriously by philosophers of

physics, the GRW interpretation posits indeterministic fundamental laws, whereas Bohmian

mechanics and most versions of the many worlds interpretation do not.
5

Notice that the claim is not that chances arise only in this case or only in cases that are relevantly

like this in some important way. For instance, this case, like the others I focus on throughout the

article, involves probabilities as they arise in a scientific setting. This is not to imply that chance

plays no role in our everyday unscientific lives. The scientific cases are merely chosen for clar-

ity—in those cases, it is easy to find agreement about the nature of the underlying laws.

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Consider, for instance, Sally’s utterance of (1):

(1) It is very likely, but not certain, that the next silver atom will be

deflected up.

One straightforward way of interpreting (1) is as a claim about Sally’s cre-

dence or degree of belief. Because she knows that it has been prepared a

certain way, Sally is confident, although not certain, that the next silver

atom will be deflected up. But according to virtually every theory of chance

(and according to many people’s pre-theoretic intuitions about the concept), it

is also possible to interpret (1) as making a stronger claim—a claim about the

chance, or the objective probability, of the next silver atom being deflected up.

This stronger claim, whatever else it amounts to, is supposed to be independ-

ent of Sally’s particular epistemic position: the chance-interpretation of (1) is

true or false in virtue of what the world is like, independent of what anyone

happens to believe about that world. If, unbeknownst to Sally, someone has

broken into her laboratory and tampered with her experimental set-up, the

chance-interpretation of (1) may turn out to be false, even though the

credence-interpretation of (1) will still be true.

It will be important to keep in mind in what follows that this familiar dis-

tinction between chance and credence is not generally taken to be exhaustive.

Consider Sally’s utterance of (2):

(2) It is very likely, but not certain, that the last silver atom was

deflected up.

Certainly we can interpret (2) as a true claim about Sally’s credences.

(Suppose, for instance, that Sally has gone to lunch and left the experiment

running so that she does not know whether the last silver atom was deflected

up.) But the description of the experimental set-up here also seems to warrant

a stronger interpretation of (2), an interpretation according to which (2) is not

a mere report of a particular person’s credence at a particular time, but a more

general claim about the credences that anyone in Sally’s epistemic position

ought to have. On this second, stronger interpretation, (2) is supposed to

convey the normative fact that anyone who knows what Sally knows about

the experimental set-up and about the results obtained so far, but is ignorant

with respect to the question of whether the last silver atom was deflected up,

ought to have a high credence in the last silver atom having been deflected up.

But whatever this normative interpretation of (2) amounts to, it is not obvi-

ously a fact about the chance of the last silver atom having been deflected up.

In particular, anyone who thinks that the past is not chancy—anyone who

thinks that the chance of any proposition that is wholly about events in the

past being true is zero or one—will think that the normative interpretation of

(2) cannot simply be a claim about chance.

Nina Emery4

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For this reason, in what follows, I will refer to the distinction we are inter-

ested in as the distinction between chance on the one hand and epistemic

probability on the other. Claims about epistemic probability include straight-

forward ascriptions of credences to a particular person at a particular time,

but they also potentially include claims like (2), when they are interpreted as

claims about what credences a person should have, under certain sorts of

conditions, even if they do not qualify as genuine claims about chance.
6

3 The Incompatibilist’s Criterion

Consider Sally’s counterpart Sally-C, who lives in a classical world where the

laws of nature include the laws of classical statistical mechanics.
7

For our

purposes, this means two important things. First, the dynamical laws that

govern the fundamental particles in Sally-C’s world are deterministic: the

micro-state of an isolated system at a time, combined with the fundamental

laws, is sufficient to determine the micro-state of that system at all other

times.
8

Second, in addition to characterizing any system in terms of its com-

plete micro-state, it is also possible to characterize it in terms of macro-phys-

ical variables like pressure, volume, and temperature, and there are robust

patterns in the macro-physical behaviour of the system. In particular, there are

certain sorts of macro-physical behaviour that people in Sally-C’s world never

observe, even though those sorts of behaviour are entirely possible, according

to the fundamental dynamical laws.

Suppose, for instance, that Sally-C runs an experiment that involves putting

an ice cube in a glass of lukewarm water and watching for an hour to see

whether the ice cube melts.
9

She knows, just by having examined the

6
Singular propositions like (1) are not the only sorts of probabilistic claims that arise in the

paradigm case. In the last section of the article, I will be particularly interested in more general

claims that are also presumably candidates for claims about chance, especially claims like: ‘Any

silver atom that was prepared in the way described above and that is about to go through

experimental set-up described above, is very likely, though not certain, to be deflected up’.
7

The way in which probabilities arise in classical statistical mechanics has received a great deal of

recent attention in the debate on deterministic chance (see Loewer [2001]; Albert [2000];

Maudlin [2007b]; and Schaffer [2007]). The case is interesting in large part because, although

we know that the laws of classical statistical mechanics are not true in our world, the way that

probabilities arise in classical statistical mechanics is remarkably similar to the way that prob-

abilities arise in Bohmian mechanics, and it is very much an open question whether Bohmian

mechanics is true in our world.
8

Another way to say the same thing: trajectories through phase space that are compatible with

the fundamental laws will never cross. (Phase space is the 6N-dimensional space in which each

point corresponds to a specification of the position and momentum of each particle in the

N-particle system.) Although there can be violations of determinism when the fundamental

laws are Newtonian, as they are in classical statistical mechanics (see Earman [1986]), these

violations occur in very specific and unusual situations. No harm comes from ignoring these

situations here.
9

I assume here and throughout the article that the system comprised the ice cube, the water, and

the glass is an isolated system that can be easily characterized in the standard macro-physical

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fundamental laws, that there are micro-states that are compatible with the

current macro-state of the ice cube and the glass that lead deterministically to

a surprising result: the ice will not melt but rather will become significantly

larger as the water in the glass around the ice cube begins to boil. However,

Sally-C has run many, many trials of this particular experiment, and every

time she has run the experiment, the ice has melted. Based on these observa-

tions, plus her background information about the fundamental laws, Sally-C

utters (3):

(3) It is very likely, but not certain, that the next ice cube will melt.
10

Certainly, there are true epistemic interpretations of (3); given the observa-

tions she has made and what she knows about the fundamental laws, Sally

presumably has (and should have) a high degree of belief in the next ice cube

melting. What about when (3) is interpreted as a claim about the chance of the

next ice cube melting?

Here is an initial reason for thinking that the chance-interpretation of

(3) must be false: if it were true, it would conflict with the chances generated

by the fundamental laws in Sally-C’s world. In particular, it seems that the

fundamental laws described above entail that the chance of any particular

ice cube melting is either zero or one.
11

The chance-interpretation of

(3) implies that the chance of the next ice cube melting is non-trivial. How

can the chance of the next ice cube melting both be either zero or one and be

non-trivial?

This sort of conflict appears worrisome, but it is the sort of thing that arises

with respect to claims about chance all the time. As Lewis ([1980]) demon-

strates, claims about chance are already widely acknowledged to be context-

dependent in at least two ways: their truth values vary depending on the world

and the time at which they are evaluated. Compare the world in which Sally’s

experimental set-up is as originally described (call it ‘!’), to a possible world

(call it ‘!1’) in which someone has broken into Sally’s laboratory and tam-

pered with the set-up by rotating the magnets by 180�.
12

Then there need not

be anything inconsistent in saying both that the chance of the next silver atom

being deflected up is very high, and that the chance of the next silver atom

variables of thermodynamics. This is not true, but no harm comes from our assuming that it is

for the purpose of the example.
10

Assume that the experimental set-up is such that there is an obvious referent for ‘the next ice

cube’.
11

Notice that my response to this reason for thinking that the chance-interpretation of (3) must be

false (which is presented in the next two paragraphs) applies regardless of how the incompatibi-

list argues for this claim.
12

To avoid unnecessary complications, assume that this was all the tampering that was done, and

thus that the tampering left a unique obvious candidate for the referent of ‘the next silver atom’

in o1 and that that candidate, on standard accounts of trans-world identity, is the same silver
atom as the referent of ‘the next silver atom’ in o.

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being deflected up is very low—it is very high in o and very low in !1.
Similarly, suppose that the tampering occurred at 10:00 pm. Then there

need not be anything inconsistent in saying both that the chance of the next

silver atom being deflected up is very high in !1, and that the chance of the

next silver atom being deflected up is very low in !1—it is very high in !1
before 10 p.m. and very low in !1 after 10 p.m.

With respect to the worry voiced above—that there is an apparent conflict

between the chances generated by the fundamental laws and the chance-inter-

pretation of (3)—the compatibilist can make a similar move. There need not

be anything inconsistent in her saying both that the chance of the next ice cube

melting is either 0 or 1 and that it is non-trivial, as long as we understand her to

be saying that the chance of the next ice cube melting is either zero or one

relative to the micro-physical facts about the system and the chance of the next

ice cube melting is non-trivial relative to the macro-physical facts about the

system. Or, to coin some new terminology: the compatibilist need only claim

that the ‘macro-chance’ of a particular ice cube melting within the hour can be

very high, even if the micro-chance of it melting is very low.
13

Given that claims about chance are already acknowledged to be context-

dependent in the two ways described by Lewis, it is up to the incompatibilist to

give some reason for thinking that they cannot be context-dependent in the

further way described above. It is up to the incompatibilist, in other words, to

provide some criterion for distinguishing between chance and mere epistemic

probability, one that classifies micro-chance in the former category, while

relegating macro-chance to the latter.

As it is usually presented in the recent literature on chance, this criterion is a

precisification of the vague but compelling thought that was introduced at the

beginning of this article. According to that initial thought, possibility is a

necessary condition for non-trivial chance: if there is a chance of something

happening, it is possible for that thing to happen and possible for that thing

not to happen.
14

Saying only that much does not get us very far, however,

13
One advantage of this terminology is that just talking about ‘the chance’ suggests that one is

using a definite description, which implies uniqueness. We often use similar constructions in

context-dependent claims. For example, ‘the weather was awful’ was true when said of this

morning, but false when said of this afternoon. A disadvantage of this terminology is that it

implies that there are two distinct types of chance. Notice that for all we’ve said here, macro--

chance is a distinctive type of chance only in the same way that chance at t1 is a distinctive type of

chance.
14

Notice that the view is not that possibility is sufficient for chance. The standard probability

calculus entails that whenever there are continuously many possible outcomes the chance of any

particular outcome is zero. So, for instance, if you are throwing a point-sized dart at a continu-

ous dart board the chance of the dart hitting the bull’s-eye (or any other point) will be zero, even

though it is possible for the dart to hit the bull’s-eye. Eagle ([2011]) argues that whether some X

can ’ provides both a necessary and a sufficient condition on the chance of X ’-ing being
non-trivial. He does so by arguing that ‘X can ’’ is not synonymous with ‘it is possible that X
’’s’. Handfield and Wilson ([forthcoming]) present a related but distinct connection between

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because whether something is possible depends on what sense of possibility we

are interested in—what facts we are holding fixed across the space of possi-

bilities that we are considering. Is it possible for Sally to attend the party?

Holding fixed all of the laws of nature, the fact that the party is on the other

side of town, and the fact that Sally is still in her office 1 second before the

party ends, the answer is ‘no’. But if we are willing to entertain possibilities in

which Sally isn’t in her office one second before the party ends, or in which she

is allowed to violate the laws of nature—by travelling faster than the speed of

light, say—then the answer may well be ‘yes’.

So the immediate question that arises with respect to the connection be-

tween chance and possibility is: what sorts of facts should we be holding fixed?

If there is non-trivial chance of Sally attending the party, then it must be

possible for her to attend the party, and for her not to attend—but possible

in what sense? Different ways of drawing the distinction between chance and

epistemic probability amount to different ways of answering this question.

Take, for example, the view that the past is not chancy—the view that the

chance of any proposition wholly about the past being true is either zero or

one. According to that view, if there is a non-trivial chance of something

happening, then it is possible for that thing to happen, and for that thing

not to happen, even when holding fixed the facts about the past.

The claim that chances are incompatible with deterministic laws can be set

out in a similar way.
15

According to the incompatibilist, if there is a non-trivial

chance of something happening, then it is possible for that thing to happen

and for that thing not to happen even when holding fixed all of the facts about

the past and the laws of nature. Or, to present the claim in the form it takes in

the recent literature on chance:

The incompatibilist’s criterion: If the chance, at world !, at time t,

of proposition p is greater than zero, then there exists a world o0

such that (i) !0 matches o in laws, (ii) !0 and ! have the same
micro-physical history up until time t, and (iii) p is true at !0.16

One of the things that is attractive about this criterion is that it seems to

capture what is distinctive about the paradigm case. It was a central feature

chance and possibility. On their view, A has some non-trivial chance of occurring if and only if

the evidence that could be acquired about A does not rule out that A occurs.
15

Incompatibilism will follow from the claim that the past is not chancy if you also think that the

laws of nature are not chancy, but the latter assumption will be problematic for anyone who

thinks that the laws supervene on particular matters of fact (i.e. for anyone who is a Humean in

Lewis’s sense). See (Lewis [1980]).
16

This principle is called the realization principle in (Schaffer [2007]) and (Glynn [2010]). It’s a

stronger version of the basic chance principle found in (Bigelow et al. [1993]). For a discussion of

why anyone who accepts the basic chance principle should also accept the realization principle,

see (Schaffer [2007], p. 124).

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of that case that all the facts about the history of Sally’s world up to and

including the moment that she utters (1), combined with the laws of nature, do

not determine whether the next silver atom will be deflected up. There are

possible worlds that match the entire history of Sally’s world up until she

utters (1) in which the next silver atom is deflected up, and possible worlds

that match the entire history of Sally’s world up until she utters (1) in which

the next silver atom is deflected down, so the incompatibilist’s criterion allows

for the chance-interpretation of (1) to be true.
17

Not so for (2) and (3). In the former case, the micro-physical history of

Sally’s world up until the point at which she utters (2) includes some fact about

whether the last silver atom was deflected up. If the last silver atom was de-

flected up, then there is no possible world that matches the history of Sally’s

world up until the point at which she utters (2) in which it was deflected down.

If the last silver atom was deflected down, then there is no possible world that

matches the history of Sally’s world up until the point at which she utters (2) in

which it was deflected up. In neither case can there be no non-trivial chance of

the last silver atom having been deflected up. So the chance-interpretation of

(2) must be false. The same will be true of all claims about the non-trivial

probability of some proposition about the past.
18

In the latter case, for any particular ice cube placed in a glass of water, the

micro-physical history of Sally-C’s world up until the point at which she utters

(3), plus the laws of nature entail that either the ice melts or that it does not

melt. If the ice is going to melt, then there is no possible world that matches the

history of Sally-C’s world up until the point at which she utters (3) in which it

does not melt. If the ice is not going to melt, then there is no possible world

that matches the history of Sally-C’s world up until the point at which she

utters (3) in which it does not melt. In neither case is the chance of the next ice

cube melting non-trivial. And since (3) says that there is some non-trivial

probability of the next ice cube melting, the chance-interpretation of (3)

must be false. The same will be true of all claims about the non-trivial prob-

ability of any proposition about a world that has deterministic fundamental

laws.
19

17
It is important that the incompatibilist’s criterion is only a necessary condition, not a sufficient

condition. You can have genuine indeterminacy without any non-trivial chances if there aren’t

the right sorts of patterns in the events. The paradigm case was set up to exhibit precisely the

right sort of pattern.
18

More carefully, claims about chance can be time-indexed in two ways. First, each claim is

indexed to a particular time of evaluation (when it is not made explicit, this is generally the

time of assertion). Call the time of evaluation te. Second, each claim about chance may be about

a particular time, call it ta. For (2), ta is before te. That is, at the time of evaluation, we would say

that the proposition to be evaluated is about the past, or about something that has already

happened. For (1) and (3), ta is after te. That is, at the time of evaluation, we would say that the

proposition to be evaluated is about the future or about something that has yet to happen.
19

One possible exception to this claim is propositions about the initial conditions of the universe.

See Loewer ([2001]) and Maudlin ([2007b]) for arguments that chances in deterministic worlds

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But notice that what the incompatibilist’s criterion says about the chance-

interpretation of (3) is due to a specific choice about what kind of past facts are

relevant. Compare the incompatibilist’s criterion presented above with the

following alternative criterion:

The alternative criterion: If the chance, at world !, at time t, of

proposition p is greater than 0, then there exists a world !0 such

that (i) !0 matches o in laws, (ii) !0 and ! have the same macro-
physical history up until time t, and (iii) p is true at !0.

Call the next ice cube that Sally-C places in a glass of water ‘N’. There

are possible worlds that match Sally-C’s world with respect to all of the

macro-physical facts up until the time she utters (3) and the laws of nature,

in which N melts. There are also possible worlds that match Sally-C’s world

with respect to all of the macro-physical facts up until the time she utters (3),

and the laws of nature, in which N doesn’t melt. According to the alternative

criterion then, the chance-interpretation of (3) may well be true.

So the incompatibilist faces the following challenge: The incompatibilist’s

criterion and the alternative criterion give different results. If we are going to

adopt one instead of the other, we need to come up with a good reason why.

We need to identify some important difference between these two criteria, a

difference that justifies our using the former instead of the latter and shows

why the former instead of the latter is the best candidate for capturing the

intuitive connection between chance and possibility. In the next section, I will

argue that this cannot be done. Given a certain well-supported assumption

about the structure of spacetime, any reason for adopting the incompatibilist’s

criterion over the alternative criterion is a reason for adopting a stronger

criterion, and that stronger criterion is implausible—it says that there are

no non-trivial chances, even in the paradigm case.

The compatibilist, as described above, however, will not face any similar

challenge. On her view, the truth value of claims about chance is not always

fixed, even after you specify a particular world and time of evaluation. They

might be true when interpreted as claims about micro-chance and false when

interpreted as claims about macro-chance, or vice versa. So the compatibilist

need not argue for the alternative criterion over the incompatibilist’s criterion

(or vice versa); she can accept both, as long as the relevant sense of ‘chance’ is

clarified in each. She will think, in other words, that if there is a non-trivial

macro-chance of something happening, then it must be possible that it hap-

pens and that it doesn’t happen, even while holding fixed all the facts about the

should be understood as probability distributions over the initial conditions of the universe.

That sort of view is not open to anyone who thinks that chance must be dynamic, as nothing

brings about the initial state of the universe.

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macro-physical history of the world; and if there is a non-trivial micro-chance

of something happening, then it must be possible that it happens and that it

doesn’t happen, even while holding fixed all the facts about the micro-physical

history of the world.

4 Against the Incompatibilist’s Criterion

Both the incompatibilist’s and the alternative criterion rely on a claim about

what is possible relative to a certain set of facts. According to the former, what

is important is whether an outcome is possible, given the set of propositions

that pick out the entire micro-physical history of the world up until t and the

actual laws of nature. According to the latter what is important is whether an

outcome is possible, given the set of propositions that pick out just the macro-

physical history of the world up until t and the actual laws of nature. In neither

case, however, is what is important whether an outcome is possible, given the

set of propositions that pick out the entire history of the world and the actual

laws of nature.

Any good reason for using the incompatibilist’s criterion instead of the

alternative criterion will therefore rely on there being an important difference

between restricting our attention to just facts about the present and the past,

on the one hand, and restricting our attention to just facts about the macro-

physical level, on the other hand. I will argue that there is no such difference.

To restrict our attention to the facts about the past and present is just as

arbitrary as restricting our attention to facts about the macro-physical world.

There is no question but that this is a surprising claim to make. Our pre-

theoretic understanding of the world around us implies that the past and the

present are importantly different from the future. But there are good reasons,

both philosophical and scientific, to think that our pre-theoretic understand-

ing is wrong.

The most straightforward reason to favour the incompatibilist’s criterion

over the alternative criterion is an ontological reason.
20

If you think the future

does not exist, then you will think that the reason we ought to take into

account the entire micro-physical history of the world, instead of just the

macrophysical history of the world, when determining whether there is a

non-trivial chance of something happening is simple: the latter leaves some-

thing out, whereas the former is complete.

To make the sort of view I have in mind explicit, assume for the moment

that there is a unique way of separating spacetime into regions of space and

20
Though this is the most straightforward reason, it is not a particularly popular one, for the rea-

sons I provide below. Dealing with it in detail here is helpful, however, as it illustrates the general

principles that are relevant in responding to other potential reasons for favouring the incompa-

tibilist’s criterion.

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instants of time. We can divide metaphysical theories of spacetime into two

camps: eternalist theories and anti-eternalist theories. According to both

eternalist and anti-eternalist theories, although the future will exist, the

future does not presently exist. The crucial difference between the two views

is that according to eternalist theories the future also exists, where ‘exists’ is

meant tenselessly. According to anti-eternalist theories, it does not.
21

There are two main lines of objection to anti-eternalist theories. The first

focuses on difficulties involved in spelling the view out in a philosophically

rigorous manner. The view that the future does not exist but will exist seems to

require that there is an objective flow of time, sometimes called objective

becoming, which gives rise to notoriously tricky philosophical questions

like: If there is objective becoming, then at the moment, the universe is smaller

than it will be in the future; but if the universe is expanding, what is it ex-

panding into? At what rate is it growing?
22

More importantly, however, it seems that anti-eternalist theories are incom-

patible with our best contemporary scientific theories. In relativistic theories

of spacetime, there is no absolute standard of simultaneity; there is no single

correct answer to the question: are events e1 and e2 simultaneous?
23

Nor is

there, in general, a single correct answer to the question: is event e1 past,

present, or future? To combine such theories with an anti-eternalist theory

(according to which only the past and present exist) would yield the result that

there is no single correct answer to the question: does event e1 exist? But to

endorse that sort of relativism about existence is absurd.
24

21
Anti-eternalist theories include presentism, according to which only the present exists, and what

is sometimes called the growing block theory, according to which only the present and the past

exist. If you are tempted to think there is not a genuine debate between the eternalist and the

anti-eternalist, or to worry that there is something suspect going on with the tenseless use of

‘exists’, it helps to draw an analogy with the debate over the existence of concrete possible

worlds. According to both the ‘actualist’ and the ‘modal realist’, merely possible worlds possibly

exist, and merely possible worlds don’t actually exist. What the two views disagree about is

whether merely possible worlds exist simpliciter.
22

For further discussion of these and other issues, see (Price [1996], Chapter 1; Markosian [1993];

Smart [1949]). A recent defence of the view that time passes can be found in (Maudlin

[2007a])—but that view, notably, is silent on the question of whether the future exists; it requires

merely that time have a unique direction. It is possible to have an eternalist theory that includes

the objective flow of time—this is usually called the moving spotlight theory. The point here is

just that eternalist theories do not require objective becoming, whereas anti-eternalist theories

do.
23

There is no privileged way, in other words, to separate four-dimensional spacetime into three

spatial dimensions and one temporal dimension. There are instead many ways of separating

regions of space from instants of time, each of which is called a ‘reference frame’. Depending on

how fast you are moving relative to other entities, different reference frames will be more or less

natural choices for you to use when making calculations or in representing various states of

affairs, but, according to these theories, no choice of reference frame is privileged.
24

See (Sider [2003], Chapter 2) for more on this objection. This is a complicated issue and deserves

more attention than I have space to give it here, especially as there are a few philosophers who

have taken the position that in light of the conflict between the anti-eternalist theories and

special relativity, we ought to revise special relativity. (Perhaps the best examples of this is

found in (Prior [1996]).) The main worry for these revisionist view is that revising special

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These considerations put us back where we started: what justifies choosing

the incompatibilist’s criterion over the alternative criterion? Both evaluate

probabilities on the basis of what is possible, relative to some restricted set

of propositions. In what sense is the set of propositions that includes the

micro-physical history of the actual world up until some particular time of

evaluation a better context against which to judge whether there is a non-

trivial chance of something happening than the set of propositions that in-

clude just the macro-physical history of the actual world up until the relevant

time? Because we live in an eternalist world, neither includes all of the prop-

ositions that are true of the actual world.

Crucially, the advocate of the incompatibilist’s criterion cannot appeal to

the simple fact that this criterion holds fixed more of the propositions that are

true of the actual world than the alternative criterion does—and is in that

sense more restrictive and thus stronger. If that is the motivation for adopting

the incompatibilist’s criterion, then, in light of the fact that we live in

an eternalist universe, surely we ought to endorse the following criterion

instead:

The eliminativist’s criterion: If the chance, at world !, at time t, of

proposition p is greater than zero, then there exists a world !0 such

that (i) !0 matches ! in laws; (ii) !0 and ! have the same complete

history through all time; and (iii) p is true at !0.

By this standard there will be no non-trivial chance of any proposition being

true in our world, whether it is about the past, present, or future. The complete

history of the world either includes that event or does not include that event,

and in either case there will be no non-trivial chance of that proposition being

true. In other words, anyone who insists on using the eliminativist’s criterion

would be adopting a distinction that counts even the probabilities in the

paradigm case as mere epistemic probabilities. At the very least, this kind of

eliminativism about chance ought to be a last resort, to be adopted only after

we are certain that there is no other way of drawing a metaphysically robust

distinction between chance and epistemic probabilities.

The ontological motivation is not the only way of motivating the incompa-

tibilist’s criterion over the alternative criterion. The argument above shows

that the problem with the latter criterion cannot be that it leaves out some

facts about the actual world. The incompatibilist’s criterion also leaves out

some facts—facts about the future. But to say that both the future and the past

relativity will involve giving up a plausible sort of simplicity assumption, namely, that if we are

choosing between two theories, both of which reproduce all the same empirical results, but only

one of which requires an extra theoretical entity (in this case a preferred reference frame), we

ought to prefer the theory that does not require the additional theoretical entity.

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exist, as the eternalist does, is not to say that the future is no different than the

past. Nor it is obvious that all facts are equally relevant when it comes to

determining the chance of a particular event occurring. So perhaps the advo-

cate of the incompatibilist’s criterion can motivate her position by arguing

that the alternative criterion, unlike the incompatibilist’s criterion, leaves out a

certain type of fact, one that makes an important difference in determining the

chances.

Anyone attempting to make such an argument needs to meet two distinct

challenges: First, she needs to identify a type of fact that is left out by the

alternative criterion but not by the incompatibilist’s criterion. Second, she

needs to establish that there is good reason for thinking that the class of salient

facts plays a distinctive and relevant metaphysical role—a role that cannot be

played by the macro-physical facts alone.

It would, for instance, be a non-starter to say simply that the incompatibi-

list’s criterion is preferable to the alternative criterion because the latter leaves

out some facts about the fundamental level. After all, the incompatibilist’s

criterion also leaves out some fundamental facts—fundamental facts about

the future. It will also not do to say only that that the incompatibilist’s cri-

terion is preferable to the alternative criterion because the incompatibilist’s

criterion leaves out some facts about the past. Saying only that much gives us

no insight into why facts about the past play a distinctive role in determining

which chances are trivial and which are not.
25

Not much more illuminating is the claim that the incompatibilist’s criterion

is preferable to the alternative criterion because the latter leaves out some of

the fixed facts. Given that the future exists, just like the past, what reason do

we have for thinking that only the past facts are fixed? One could just stipulate

that the term ‘fixed’ includes all and only the past facts (and, perhaps, facts

derivable from the past facts using the laws of nature); but if this is just a

stipulation, then, again, why think that this group of facts plays an important

and relevant metaphysical role, one that cannot be played by the macro-phys-

ical past facts alone?

A more promising move is to appeal to the temporal asymmetry in caus-

ation. It is common to assume, either implicitly or explicitly, that causes must

precede their effects. Assuming that causes always precede their effects, if you

25
A similar point applies to the distinction, found in (Barnes and Cameron [2009]), between

possible worlds and possible futures. On their view a possible future is just a world that matches

the actual world with respect to all of the facts about the past. The introduction of this termin-

ology allows us to identify a group of facts that is captured by the incompatibilist’s criterion but

not the alternative criterion (the latter leaves out some facts that are included in all of the

possible futures). But this new piece of terminology alone does not give us any good reason

for thinking that the facts that are held in common across all possible futures plays some

important metaphysical role with respect to fixing the chances, a role that cannot be played

by, for instance, the facts that are held in common across all macro-possible futures.

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take into account the entire micro-physical history leading up to some event,

you will have taken into account all of the potential causes of that event. So,

according to this way of thinking, the incompatibilist’s criterion is preferable

to the alternative criterion because the latter leaves out some of the potential

causes of relevant events.
26

The first thing to say about this motivation is that once we fully internalize

the discussion above about the implausibility of anti-eternalist theories, the

temporal asymmetry of causation (the claim that causes always precede their

effects) may be open to doubt. At the very least, given the implausibility of

anti-eternalist theories, the temporal asymmetry of causation is something

that we learn about the world, not something we know a priori. So it

cannot be, for instance, that the incompatibilist’s criterion is preferable to

the alternative criterion because the latter leaves out some facts that, for all

we know a priori, are causes of the relevant effect.

In any case, it is not clear why, when determining whether there is a non-

trivial chance of something happening, we ought to be interested in the po-

tential causes of the relevant events, as opposed to the actual causes. It is by no

means obvious that the actual causes of any particular event will always in-

clude facts about the micro-physical level.

To put the point another way, the advocate of the view described above

does not seem to be arguing for the incompatibilist’s criterion so much as the

causal criterion:

The causal criterion: If the chance, at world !, at time t, of prop-

osition p is greater than zero, then there exists a world !0 such that

(i) !0 matches ! in laws; (ii) !0 and ! match in terms of all of the

causes of events that appear in p; and (iii) p is true at !0.

Unless one is an arch-reductionist about causation—that is, unless one thinks

that one must always hold all of the facts about the micro-physical history of

an event fixed to hold the causal facts about that event fixed—it is by no means

obvious that the causal criterion and the incompatibilist’s criterion amount to

the same thing. That sort of arch-reductionism is a substantial commitment to

take on to save the incompatibilist’s criterion, especially before we have inves-

tigated other possible ways of drawing the distinction between chance and

epistemic probability. For one thing, such arch-reductionism conflicts with

any theory of causation according to which causes are required to be com-

mensurate or proportional to their effects.
27

For another, it seems to be part of

26
Thanks to Brad Skow for suggesting this line of thinking.

27
On such theories what causes an ice cube in a glass of water to melt will be the initial macro-state

of the system, not the initial micro-state, as the vast, vast majority of the information contained

in the initial micro-state could have been different and the macro-physical behaviour of the ice

remained exactly the same. See, for instance, (Yablo [1992]). A couple of points here: First, it is

possible to read Yablo as giving not a metaphysical account of causation so much as a pragmatic

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our pre-theoretic understanding of causation that there are patterns of causal

dependence at various levels of description, including the macro-physical level

(pool balls cause one another to accelerate, snowstorms cause car accidents,

hunger causes people to head for the kitchen) and that when such patterns

occur, it is possible for the micro-physical facts to vary without changing the

causal facts. (Fundamental particle p might have had an ever-so-slightly dif-

ferent momentum, for instance, and yet the cue ball would still have caused the

eight ball to go into the pocket.)

There is one final sort of motivation for the incompatibilist’s criterion that I

will consider. Maybe taking into account the complete micro-physical history

up until some relevant time is the best we can do, not because of some inde-

pendent metaphysical feature of the world, like that the future doesn’t exist or

that causes always precede their effects but because of some fact about us.

Perhaps the reason why we should use the incompatibilist’s criterion instead

of the alternative criterion to pick out the objective probabilities is because we

have a different sort of access to micro-physical facts than we do to facts about

the future.

The problem with this motivation is that when you focus on the sort of

micro-physical facts that actually play a role in theories like classical statistical

mechanics, it is by no means obvious that we do have better access to these

micro-physical facts than the kind of access we have to facts about the future.

Even assuming that a glass of water with ice in it is an isolated system, in order

to determine for certain whether the ice will melt, we would need to know the

exact position and momentum of each of the fundamental particles in the

glass, the water, and the ice at a single time.
28

Even with the best-equipped

of physics laboratories at our disposal, this is not the sort of information that

we actually have access to.

It is easy to imagine someone protesting that what is important is not the

sort of information we actually have access to, but instead the sort of infor-

mation that we could in principle have access to.
29

But in what sense of ‘in

principle’ can we in principle come to know the exact micro-state of the system

account of what causal factors it is contextually appropriate to cite. Insofar as his article ad-

vocates the kind of view I have in mind here, he should not be read that way. Second, Yablo

distinguishes between multiple causal notions, in particular, he distinguishes between causation

and causal relevance, and only the former must be commensurate in the sense described above.

In other words, the initial micro-state might still be causally relevant to the final macro-state

even if it isn’t the cause of the final macro-state. This raises the prospect that the causal motiv-

ation discussed in this section might be resurrected as the causal relevance motivation. But

again, the advocate of such a view faces two challenges: first, show that facts about the future

are not causally relevant, and second provide some argument for the claim that chances always

depend importantly on everything that is causally relevant as opposed to only on those things

that are actual causes.
28

The problem is much, much worse, if the only truly isolated system is the entire universe.
29

See (Ismael [2012], p. 425) for a defense of this position.

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but cannot, even in principle, know what is going to happen in the future

(except to the extent that it is determined by some past or present events)?

There are physical facts about us, about what sorts of beings we are and how

we are situated in spacetime that make it impossible for us to build a measur-

ing device that delivers regular information about the future, and there are

physical facts about us and how we are situated at the macro-physical level

that make it impossible for us to build a measuring device that delivers regular

information about the exact micro-state of everyday systems. If we are

allowed to vary the former sort of fact, why shouldn’t we be allowed to

vary the latter? Put another way, it is of course possible that a different sort

of being than us—the sort of demon that Maxwell once conjectured—would

be able to determine the exact micro-state. But unless there is some important

metaphysical distinction between the past and the present on the one hand,

and the future on the other, then there is no obvious reason why there could

not also be an analogous being—an Einsteinian demon, say—who would be

able to determine the exact future state of the cup.

5 The Explanatory Criterion

The arguments in Section 4 leave us with two main options.
30

First, we can

become eliminativists about chance. According to this way of thinking, rela-

tivistic theories have not only taught us something important about spacetime

(that we live in an eternalist world) but also about chance (that there are no

non-trivial chances in our world).

Second, we can reject the incompatibilist’s criterion and take seriously the

suggestion made in Section 3 that chances are relative not only to a world and

a time of evaluation but also to a specification of whether they are being

evaluated at the macro-physical or micro-physical level. Relative to the

micro-physical facts, the chance of any event in a world where the laws are

deterministic is either zero or one. Relative to the macro-physical facts, there

may be non-trivial chances in such worlds. There is no reason we cannot

endorse both of these claims, just as there is no reason we cannot endorse

apparently conflicting claims about chance when those claims are interpreted

as claims about the chance as evaluated at different times or in different

worlds. The connection between chance and possibility is captured trivially

by allowing for many different connections between different types of chance

and corresponding types of possibility, as described at the end of Section 3.

If we are willing to relativize chance to a specification of macro- and micro-

levels, the question arises: ought we to relativize it in other ways? In addition

30
A third option being to adopt arch-reductionism about causation in order to save the incom-

patibilist’s criterion.

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to micro-chance and macro-chance, is there chemical chance and biological

chance? And if so, why stop there? Why not think that for every set of facts,

there is a probability function corresponding to the chance relative to those

facts. Why not think, for instance, that in the case where someone tampered

with Sally’s experimental set-up in the way described in Section 2, the chance

interpretation of (1) is still true as long as it is interpreted as a claim about the

chance of the next silver atom being deflected up relative to what Sally knows.

The chance of the next silver atom going down may be very low relative to the

facts about how the experimental set-up is arranged, but why should that

preclude the chance of the next silver atom going up from being very high

relative to what Sally knows about that set-up? And if we are willing to adopt

this last suggestion, if we are willing to say that so-called ‘Sally-chance’ is just

as deserving of the name ‘chance’ as macro-chance or micro-chance, it is no

longer clear that there is an important distinction between chance and epi-

stemic probability at all.
31

Here is another way of putting the worry. If the incompatibilist’s criterion

was our only hope for making a metaphysically robust distinction between

chance and epistemic probability, then the two options left to us at the end of

Section 4 present us with two horns of a dilemma: either we eliminate chance

or run the risk of trivializing it.

It is important to establish, therefore, that the incompatibilist’s criterion is

not our only hope for making a metaphysically robust distinction between

chance and epistemic probability, and thus that the dilemma above is one we

can avoid. As it happens, there is another distinctive feature of the paradigm

case that we have yet to consider: the explanatory role played by the prob-

abilities that arise in that case.

Look again at the paradigm case. In that scenario, Sally observed a certain

pattern in the results of her experiment: most (but not all) of the silver atoms

31
The point here is not that there is no way of resisting these questions, only that they pose a prima

facie worry. Glynn ([2010]), for instance, argues that chance should be relativized to each sci-

entific level, and presumably no further. The success of that argument of course depends on

developing a clear understanding of what counts as a scientific level. If such levels are just

defined as sets of facts that are explanatorily closed, then the distinction between chance and

epistemic probabilities on that view will likely end up being the same as the distinction I suggest

below.

Handfield and Wilson ([forthcoming]) go a step further. On their view, chance is relativized to

an ‘evidence base’—a set of propositions that is determined by the context. Insofar as this view is

meant to provide a distinction between chance and mere epistemic probability, all of the work is

in determining which evidence bases generate genuine chance functions. The authors do not

attempt to give necessary and sufficient conditions for when a particular evidence basis can

generate genuine chance-functions (provided the right context), but they do argue that overly

gerrymandered evidence bases do not generate genuine chance-functions. Neither do overly

parochial evidence bases (evidence bases that include only propositions about a localized

region of spacetime. The evidence basis that consists of all of the facts about the cards visible

to a poker player according to the rules of the game does, however, generate a genuine chance-

function.

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were deflected up. On the basis of the frequency of the silver atoms being

deflected up, she asserted (1), and we concluded that the chance-interpretation

of (1) was true
32

:

(4) The chance that the next silver atom will be deflected up is very high.

But (4) is not the only sort of chance claim that Sally seems warranted to make

in that scenario. In particular, (4) seems to follow from a more general claim

which, given the observations she has made, is liable to be part of Sally’s best

theory about the experimental set-up:

(5) The chance of any silver atom in the experiment being deflected up is

very high.

And (5) demonstrates that in addition to claims about frequency

providing evidence for claims about chance, claims about chance can also,

at least sometimes, explain claims about frequencies. In particular, (5)

explains (6):

(6) Most of the silver atoms in the experiment have been deflected up.

That (5) explains (6) may not immediately obvious, but the argument for it

is straightforward. First, notice that if (5) does not explain (6), then nothing

does. It is part of the paradigm case that there is no feature such that all and

only silver atoms that are deflected up have that feature before they are sent

through the magnets. What else, then, could explain the fact that most of the

silver atoms sent through the experiment are deflected up? What other sort of

fact could be provided as answer to the question, ‘Why have most of the silver

atoms in the experiment been deflected up?’ Second, notice that if nothing

explains (6), then we have no reason for expecting the pattern described in (6)

to continue. If it is just a massive coincidence that most of the silver atoms sent

through the experiment have been deflected up, then we should not expect the

next silver atom to be deflected up. But, we do expect the next silver atom to be

deflected up, and most of the silver atoms after that. So we expect the pattern

described in (6) to continue. So something explains (6). And the only possible

explanation is (5).
33

32
This is not to say that frequencies are the only evidence we have for chance, nor is it to say that

frequencies cannot sometimes provide misleading evidence about chances. It is a crucial part of

our concept of chance that the relative frequency of some event can vary significantly from the

chance of that event, though it is unlikely to do so.
33

Crucially, I am not claiming that all facts cry out for explanation in the sense that (6) does.

Contrast, for instance, the fact that most silver atoms are deflected up with the fact that one

particular silver atom was deflected up. The argument above shows that we cannot leave the

former fact unexplained, but does not address the latter fact.

The discussion here illustrates the complicated relationship between prediction and explan-

ation in scientific theories. Predictions answer questions of the form ‘Will E occur?’, whereas

explanations answer questions of the form ‘Why does E occur?’ But part of being a good

explanation is also to underwrite certain sorts of predictions. A good answer to the question

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The fact that (5) explains (6), combined with the fact that (5) describes

what is supposed to be a paradigm case of chance, suggests the following

criterion:

The explanatory criterion: Suppose that events of type E are events

that are independent of anyone’s belief state. If (I) explains (II),

then the chance-interpretation of (I) is true:

(i) The probability of any event of type E is very high.

(ii) The relative frequency of events of type E is very high.
34

It is important to emphasize that the explanatory criterion as set out here does

not obviously capture everything there is to say about the explanatory role

that chances play. It is silent, for instance, on the question of whether the

chance-interpretation of (I) also explains any particular instance of an event of

type E.
35

But even presented in this limited form, the explanatory criterion keeps us

from sliding toward the kind of trivialization presented earlier in this section.

Consider the case where someone has tampered with Sally’s experimental set-

up in the paradigm case by rotating the magnets by 180�, and suppose that (as

is possible, because the laws in that case are indeterministic) most of the silver

atoms continue to be deflected up. In that case there is an obvious sense in

which it is still very likely for any next silver atom to be deflected up—for all

that Sally knows, it is still very likely. But that fact does not explain the fact

that most of the silver atoms continue to be deflected up. What Sally knows or

does not know does not have any relevance at all to the behaviour of the silver

‘Why does E occur?’ should provide you with the ability to make various predictions about the

conditions under which E-type events will occur in the future.
34

The explanatory criterion obviously undermines a straightforward actual frequency interpret-

ation of chance, according to which the chance of some outcome, O, occurring as a result of

some set-up, E, is just the actual relative frequency with which outcomes like O result from

set-ups like E. If (I) is just equivalent to (II), then (I) cannot explain (II). This is no real detriment

to the criterion, as simple actual frequentism already faces significant objections. Perhaps the

most straightforward and compelling objection to the view is the only already mentioned in

Footnote 31 above: it seems possible for the actual relative frequency of some type of event to

differ from the chance of that type of event. For instance, it seems possible, though unlikely, that

a six could come up far more often than one sixth of the time, when rolling a fair die. This

possibility is part of what we are trying to convey when we claim that the die is fair. For a

comprehensive rehearsal of the arguments against actual frequentism see (Hajek [1996]). Hajek

takes the fact that actual frequentists cannot account for the explanatory role that chances play

as an argument against actual frequentism.

It is not obvious that the worry extends straightforwardly to more sophisticated versions of

frequentism, like hypothetical frequentism (according to which probabilities are just the relative

frequency that would occur if the experiment in question were repeated infinitely many times) or

David Lewis’s Humean theory of chance. Although it is a standard objection to Humean

theories of laws that they do not explain their instances, Humeans have argued that laws, on

their view, can still play an important explanatory role. See (van Fraassen [1989], pp. 48-51).
35

A claim that is made in (Albert [2000]) and (Loewer [2001]). It is also silent on the question of

whether very unlikely relative frequencies have any explanation.

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atoms. So the explanatory criterion does not warrant the claim that the chance

any silver atom being deflected up is still very high.

The criterion also apparently classifies at least some probabilities that arise

in worlds where the fundamental laws are deterministic as chances. Consider

again the world that Sally-C lives in, where the laws include the laws of clas-

sical statistical mechanics. In that situation, a fact about frequencies (that all

of the ice cubes Sally has observed floating in glasses of lukewarm water have

melted within the hour) plus a fact about the fundamental laws (that it is

possible for ice cubes in lukewarm water not to melt, but instead get bigger,

while the rest of the water around them boils) prompted Sally to assert:

(7) It is very likely, though not certain, that the next ice cube will melt.

And presumably also warrants her to assert:

(8) It is very likely, though not certain, that any ice cube in the experi-

ment will melt.

According to the explanatory criterion, if (8) explains (9),

(9) Most of the ice cubes in the experiment have melted,
36

then the chance-interpretation of (8) is true.

One initial reason for thinking that (8) does explain (9) is that exactly this

sort of explanatory relationship is part of the standard presentation of clas-

sical statistical mechanics. After all, it is a central part of that theory that the

fundamental laws allow for all sorts of strange behaviour that we never ob-

serve: ice cubes placed in lukewarm water can get bigger, broken coffee mugs

can knit themselves back together and hop up off the floor, a banana left on

the kitchen counter can get progressively less ripe, and so on. Which raises the

question: why don’t we ever observe such strange behaviour? The standard

answer to that question is: although that behaviour is possible, it is extremely

unlikely.
37

36
Presumably, if (8) explains (9), then (8) can also explain the stronger fact that all of the ice cube

in the experiment have melted, as long as we assume that the probability described in (8) is high

enough relative to the number of times the experiment has been run.
37

See, for example, (Sklar [1993]; Albert [2000]; Loewer [2001]). Scientists often say something

stronger: that on any particular occasion when ice melts, (8) explains why it melted. See, for

example, (Albert [2000], pp. 64-5). A related claim is that only the explanatory power of claims

like (8) can explain the historical facts regarding the adoption of statistical mechanics instead of

alternative theories in the second half of the nineteenth century. For more, see (Strevens [2000],

pp. 12-4), which is a concise version of the account found in (Brush [1983]).

One of the major projects in the foundations of statistical mechanics is to try to spell out what

is going on at the fundamental level in virtue of which (8) is true. An overview of such attempts is

found in (Uffink [2007]) and further discussion can be found in (Leeds [2003]). The attempt that

is probably currently most popular among philosophers (found in Albert [2000]), begins with the

observation that for a number of relatively simple systems, if you take phase space (the 6N-

dimensional space in which three dimensions correspond to the position of each particle in the

system and three dimensions correspond to the momentum of each particle in the system) and

you (i) place a uniform, normalized measure over any region that corresponds to a macro-state

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Unlike in the paradigm case, there are alternative explanations for (9). In

particular, one might try to explain (9) by citing only the initial micro-state of

every observed ice cube in a glass of water, and the fundamental laws,

as in (10):

(10) Every ice-cube-and-glass system observed so far started off in a

normal micro-state (a state that leads deterministically to the ice

melting within an hour).
38

The recent literature on explanation contains a number of arguments for the

conclusion that (8) is a better explanation of (9) than (10) is,
39

but notice that

the explanatory criterion as stated above does not require any such argument.

The criterion does not require that (8) must be the best explanation of (9), only

that (8) be an explanation of (9), and there is good reason to think that these

are two distinct conditions. Maintaining a robust distinction between the

claim that p1 explains p2, on the one hand, and the claim that p1 is the best

explanation of p2, on the other, is the best way of accommodating the appar-

ent fact that what counts as the best explanation of something varies consid-

erably with pragmatic features of the context without rendering the notion of

explanation a wholly pragmatic one. It is the best way, in other words, to

allow that although there may be not a single context-independent fact about

of the system and (ii) assume that the universe as a whole started off in a low-entropy state, then

the measure of the normal micro-states (states that leads deterministically to thermodynamically

acceptable behaviour) within that region will be one, and the measure of abnormal micro-states

will be zero. Advocates of this view suggest that we assume that this result is true for all systems

and interpret the relevant measure over phase space as a probability measure.

It is possible for someone to claim that this fact about the relative standard measure of

normal to abnormal micro-states explains the fact that most ice cubes melt directly and non-

probabilistically, but the advocate of such a view faces an significant challenge especially given

that there is no one unique or uniquely natural way of putting a measure over phase space. (As

discussed in, for example, (Uffink [2004], Section 4.1), a uniform, normalized measure placed

over phase space yields different results if phase space is parameterized in terms of position and

energy, instead of position and momentum.)
38

I present (10) here because it is taken to be the standard micro-level explanation of (9). (See

Woodward [2005]; Strevens [2008]; and Weslake [2010].) But there are good reasons for thinking

that (10) doesn’t explain (9) at all. For all (10) says, it could have been mere coincidence that

each ice-cube-and-glass system started off in exactly the micro-state that it did. Given that (10)

leaves it open that (9) is the result of that sort of coincidence, it gives us no reason at all for

expecting (9) to continue. But, as with the fundamental-level explanations described above, it

seems that part of what we are looking for when we are looking for an explanation of robust

patterns like that documented in (9) is an explanation of not only why we have observed that

pattern but also why we expect that pattern to continue. One way to avoid this worry is to

replace (10) with the following claim: the universe as a whole started in micro-state m, and m,

combined with the laws of nature, entails that all (or almost all) ice cubes placed in glasses of

water begin in a normal micro-state. For reasons given in, for example, (Callender [2004]), it is

plausible to think that claims about the initial state of the universe do not require further

explanation.
39

Again, see (Woodward [2005]; Strevens [2008]; Weslake [2010]).

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what the best explanation is, there are still objective facts about what sorts of

facts are explanations at all.
40

So the mere existence of an alternative explanation, like (10), does not

undermine the claim that (8) explains (9). What would suffice to undermine

that claim would be a theory of explanation that explicitly ruled out (8) as an

explanation of (9). One such theory is a deductive-nomological theory of ex-

planation, according to which p1 explains p2 if and only if p1, combined with

some law of nature, entails p2. Another such theory is one that combines the

claim that all explanation is causal explanation with the kind of arch-reduc-

tionism about causation that was described in Section 4 (the view that caus-

ation always occurs at the fundamental level). But there are serious objections

to both of these positions, objections which, at the very least make them

initially unattractive.
41

It is up to the person who endorses one of these the-

ories to provide a convincing argument for them. Until they do, the explana-

tory criterion suggests that we ought to be compatibilists about chance and

deterministic laws.

6 Conclusion

Non-trivial chances arise even in worlds where the fundamental laws are de-

terministic. This kind of compatibilism—compatibilism about chance and

deterministic laws—is not something that can only be established after adopt-

ing a detailed and controversial metaphysics of chance. Instead, it follows

from two minimal assumptions: that the case presented in Section 2 is a para-

digm case of chance and that we live in an eternalist world. To resist the

argument from these minimal assumptions to compatibilism requires adopt-

ing substantial metaphysical commitments, the sorts of commitments that

most of us would like to avoid if we could.

Rejecting the incompatibilist’s constraint on chance does not mean, how-

ever, that there is no way of drawing a robust distinction between chance and

epistemic probability. Careful attention to the paradigm case in Section 2

shows that chances play a distinctive explanatory role in that case: the high

chance of a certain type of event happening explains the high relative fre-

quency with which that type of event happens. More work needs to be done

to set out the details of this explanatory role and to understand how it

40
See (Sober [2010]) for an example of someone who rejects any important connection between

chance and explanation because he is concerned about pragmatic factors in the latter. Sober

does not distinguish, in that article, between the question of whether p1 explains p2 and the

question of whether p1 is the best explanation of p2.
41

Even Hempel ([1965]) (the locus classicus for a defence of deductive-nomological explanation)

allows for forms of statistical explanation. Salmon ([1989]) includes a review of well-known

counterexamples to the deductive-nomological view. See Sober ([1983]) for an argument that not

all explanation is causal explanation.

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generalizes to other cases, but even in the minimal form in which it is presented

here, this explanatory criterion apparently allows for non-trivial chances in

worlds where the fundamental laws are deterministic.

Compatibilism about chance has often been assumed to be a substantive

and surprising result of certain metaphysical theories of chance. As the above

arguments demonstrate, compatibilism ought to be the default view, the start-

ing place from which our metaphysics of chance begins.

Acknowledgements

Thanks to the many people who provided helpful feedback on earlier drafts,

especially Ned Hall, Brad Skow, Bob Stalnaker, Caspar Hare, Roger White,

Jim Binkoski, Alejandro Perez Carballo, and Paulina Sliwa.

Department of Philosophy

Brown University

Providence, RI 02912, USA

nina_emery@brown.edu

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